 Hello friends in this question. We are going to solve this equation. This question was asked in CBC 2013 So this is a board paper question. So let us see how we can solve this question This question belongs to the topic of quadratic equation. How do you know it? There are surrogate methods of finding that out first of all only one variable is involved and secondly This is not in numerator. That means it's clearly not a linear equation case so the other Equation which you have studied in 10th board is quadratic Equation, so we will be focusing mostly on problem-solving skills and not going into the basic details of The theory here, so okay, how do we solve such questions? So first of all you'll have to take the LCM of The left hand side why because both the X are in the denominator. So let us take the LCM So if you see LCM of any two factors or any two monomial will be nothing, but just multiply them together So and then if X minus 2 I have already taken so First of all write whatever is the numerator over here So one is numerator then multiply this by a quantity which you obtain by dividing this whole denominator by First numerator. So if I divide the entire Here the new denominator by this X minus 2 I'll get X minus 1 so hence I'll write X minus 1 then what you do then you simply add This to here and then again repeat the process. So denominator divided by this denominator So LCM denominator divided by this denominator. You'll get X minus 2 So hence it is X minus 2 and Rest you write 6 upon X Okay, now we'll have we'll just simplify So hence on the numerator in the left hand side, you'll get X minus 1 then plus 2 X minus 4 and In the denominator, you will be getting X square minus X times minus 1 is minus X then minus 2 times X is minus 2 X and Hit it is plus 2 Plus 2 and this is equal to 6 upon X. So just by expanding Now we'll simplify further and then we'll operate or we will do cross multiplication. So simplify further So X plus 2 X is 3 X and minus 1 minus 4 is minus 5 and then divide by X squared minus 3 X Plus 2 and this is equal to 6 upon X Now we will cross multiply to get what? X times 3 X. So this X This X goes and multiplies with this and this 6 goes and multiplies with this So if you see you will get 3 X so X into 3 X minus 5 This must be equal to 6 times 6 times X squared minus 3 X Plus 2 right now. Let us again simplify by expanding. So 3 X into X is 3 X square minus 5 X is equal to 6 X squared minus 18 X Plus 12 Okay, you need to be very very careful while doing the arithmetic. Okay, 6 X square minus 6 3 the 18 X plus 12 Okay, now what you will take everything on one side right and equate it to 0 So if you see 6 X square is more than 3 X square here So we'll take everything on the right-hand side and then you can write this as This is 0 and you can write 6 X square and then take this 3 X square here So 3 X square minus then minus 18 X was already here and then you when you take minus 5 X on the right-hand side It will look it will become 5 X and then plus 12 Right this is equated to 0 now you can rearrange it bring everything on the left-hand side You'll get 6 minus 3 that is 3 X squared 3 X squared minus 18 plus 5 is minus 13 X and then plus 12 Now mind you the question was to solve this by factorizing it So you cannot to use the quadratic formula. So how to factorize you need you know you have to go by splitting the middle term split split the middle term Split the Middle term how to split so if you see what is the coefficient here so splitting the mid-middle term say so a is equal to 3 here B is equal to minus 13 and C is equal to plus 12. Isn't it? Now first find out AC A times C how much is it 3 into 12 36? Okay, now you have to you have to you know break B Break B into two parts B1 and B2 such that B1 plus B2 should be equal to B and B1 into B2 must be equal to AC Okay, so hence now another way of looking at it is if you see if this sign You know AC is positive and B is negative Okay, then you have to break this From you have to break it internally that means both the parts will be lesser than 13 If it were minus 12 here and it was plus 13 here Then you have to break this in externally that means minus 13 has to be broken such that it is difference of two Numbers again, so it is plus here and here minus right so 3 into plus 12 is 36 36 and it is minus 13 So I know AC is positive and B is negative. So I have to break B from within Okay, so this B must be broken down or split it into two terms such that some is 13 and product is 36 Okay, so if you if you see other way of doing it is let's try and factorize Let's break 36 into pairs of factors. So 36 is 1 into 36 then 2 into 18 then 3 into 12 and then 4 into 9 and then 6 into 6 And then it repeats. Yeah, it repeats So it will become like next is 9 into 4 and all that so if you clearly see 13 is some of 9 plus 4 isn't it and I need to forward 36 anyway, so hence my two terms are 3x square minus 9x minus 4x plus 12 Equals I have purposefully written minus 9x first not minus 4x because I want to again extract some common factor So if you see this is 3x into x minus 3 Minus 4 common x minus 3 Equals 0 once you get one factor you can clear, you know very easily write this this factor here and extract the common factor Okay, so hence it is x minus 3 times 3x minus 4 equals 0 Okay, how is this possible only when x minus 3 is equal to 0 or x equals to 4 by 3 So these are this is a solution. So let's check whether it is really You know, you should also check The solution once you get it you should also check whether it is correct or not as always a good practice So x equals so when x equals to let's put x equals to 3 and check whether the solution is correct So this is 1 upon 3 minus 2 which is 1 plus 2 upon 3 minus 1 which is 2 2 so hence you will get Sorry 1 plus 1 is equal to 2 in the LHS in the RHS You will get 6 by 3 which is 2 so hence It satisfies Isn't it? Let's check the other solution as well. So I am checking it here Okay, so let's say if x equals to What 4 by 3 so hence it will become LHS is equal to 1 upon 4 by 3 minus 2 plus 2 upon 4 by 3 minus 1 Which is equal to if you check it is nothing but 3 into 2 6 so it is minus 3 by 2 minus 3 by 2 plus it is 6 6 upon Yeah, 6 upon 1 right so 3 times. Yeah, so 4 minus 3 is 1 1 by 3 So this is 6 upon 1. Yes, which is nothing but equal to 4.5. Isn't it 4.5 or 9 by 9 by 2 This is LHS. What is RHS? Let's check RHS is 6 upon 4 into 3 now So 18 by 4 which is 9 by 2 so hence both the solutions are Satisfying this equation so hence our solution was correct so x equals to 3 and x equals to 4 by 3 both are solution to this equation